Bulletin of Mathematical Biology

, Volume 76, Issue 1, pp 184–200 | Cite as

Optimization and Phenotype Allocation

  • Jürgen Jost
  • Ying WangEmail author
Original Article


We study the phenotype allocation problem for the stochastic evolution of a multitype population in a random environment. Our underlying model is a multitype Galton–Watson branching process in a random environment. In the multitype branching model, different types denote different phenotypes of offspring, and offspring distributions denote the allocation strategies. Two possible optimization targets are considered: the long-term growth rate of the population conditioned on nonextinction, and the extinction probability of the lineage. In a simple and biologically motivated case, we derive an explicit formula for the long-term growth rate using the random Perron–Frobenius theorem, and we give an approximation to the extinction probability by a method similar to that developed by Wilkinson. Then we obtain the optimal strategies that maximize the long-term growth rate or minimize the approximate extinction probability, respectively, in a numerical example. It turns out that different optimality criteria can lead to different strategies.


Branching process Random environment Optimization Phenotype allocation 



We would like to thank Professor Rüdiger Frey for helpful discussions and comments. We would like to thank the referees for their thoughtful comments and suggestions, especially for drawing our attention to Brennan and Lo (2011).


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Copyright information

© Society for Mathematical Biology 2013

Authors and Affiliations

  1. 1.Max-Planck-Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Department of Mathematics and Computer ScienceLeipzig UniversityLeipzigGermany
  3. 3.Santa Fe Institute for the Sciences of ComplexitySanta FeUSA
  4. 4.Warwick Systems Biology CentreThe University of WarwickCoventryUK

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